Naturalité des extensions minimales du secteur scalaire du Modèle Standard
On commence avec l'interrogation de James D. Wells sur un risque générique d'instabilité par prolifération de bosons de Higgs, sa réflexion se place dans la vision orthodoxe de la théorie quantique des champs sur l'espace-temps ordinaire:
Let us ... begin with ... just one additional scalar Φ that has no charges under the Standard Model gauge symmetries. Since |Φ|2 is gauge invariant and Lorentz invariant there is no prohibition to coupling it with the SM Higgs boson H at the renormalizable level. The resulting scalar potential is
V = −1/2(µH|H|2 − μΦ|Φ|2)+ 1/4(λH|H|4+2λHΦ|H|2|Φ|2+ λΦ|Φ|4). (2)
Assuming <H> = v and <Φ> = w, the minimization conditions for this potential are
−µH2 + λHΦ w2 + λH v2 = 0 and − µΦ2+λHΦ v2 + λΦ w2 = 0. (3)
These two equations must be satisﬁed to be at the stable minimum of the potential. If we assume all dimensionless couplings are O(1) and v2≪ w2∼ µΦ2, we have a serious problem with eq. 3. There is no reason to discount the prospect of even many condensing scalars with vacuum expectation values as high as the Planck scale, 1018 GeV, which is sixteen orders of magnitude higher than the weak scale mweak, but even just this one extra ﬁeld is destabilizing. Somehow the large µH2− λHΦ w2 ﬁrst two terms in the ﬁrst minimization equation above must cancel each other to a large ﬁne-tuned degree in order to match in magnitude the much smaller λH v2 term so that the minimization condition is satisﬁed. There are only two solutions to this problem. One, we accept a serious ﬁne-tuning of the parameters such that this cancelation occurs. Or, assume that for some reason the mixing λHΦ between the Higgs and any other condensing scalar is small so that every term of that ﬁrst equation is of the same order O(mH2). The mixing has to be at least as small as λHΦ ∼ v2/w2. There are strong arguments against both solutions to this proliferation problem. And as alluded to above, the problem gets much worse as the number of condensing scalars increases. The ﬁrst solution assumes an accidental ﬁne-tuned cancellation among terms that is hard to imagine in even just one equation. However, if we had n scalars then there would be n such minimization equations, all requiring similarly spectacular ﬁne-tuned cancellations. The small mixing solution is less than desirable also, because if there are n such scalars then we have to assume that there is at least the same small mixing for every one of them. This is no longer accidental but systematic, and so must involve a principle. This principle is unknown from the point of view of the SM and thus is not satisfactory unless “new physics” is invoked.
James D. Wells, Naturalness and the Higgs Boson Proliferation Instability Problem, 10 avril 2014
We know that the quartic couplings λh , λhσ and λσ all run, and since they are dimensionless we can trust their running and obtain their value at low scale (Λ = MZ) from the initial condition at uniﬁcation scale Λunif. On the other hand, the mass terms give quadratic divergences, and their running cannot be trusted. We write the potential in the form
V = − 1/2 (µ2h2 + ν2σ2) + 1/4 (λh h4 + 2λhσ h2σ2 + λσ σ4). The minimum is obtained for <h2>= v2, <σ2>= w2 where −µ2 +λhv2 + λhσ w2= 0 and −ν2 + λhσ v2+ λσw2 = 0.
We determine the mass terms µ2 and v2 at low scale using this equation so that v2 is of order 102 Gev and w2 is of order 1011 Gev. We will not worry about this ﬁne tuning, which is related to the problem of quadratic divergencies, but note that the dilaton ﬁeld  which replaces the scale Λ could be used to technically solve this problem as it connects the diﬀerent subalgebras of the discrete space. We use the approximation v2 ≪ w2 which is also made use of to get the see-saw mechanism.
Ali H. Chamseddine and Alain Connes, Resilience of the Spectral Standard Model, 27 Août 2012
Remarque: les notations adoptées par James D. Wells ont été partiellement modifiées pour être harmonisées avec celles de Chamseddine et Connes afin de faciliter la comparaison et souligner la similarité des discussions et des hypothèses sur les constantes de couplage entre les deux champs scalaires.
Quelques détails supplémentaires sur un boson scalaire singulet de très grande masse couplé au boson de Higgs à 126 GeV
...a tiny mixing of the Higgs with a heavy singlet can make the electroweak vacuum completely stable... as the singlet VEV increases, the Higgs mass–coupling relation recieves a tree level contribution which does not vanish in the zero–mixing/heavy–singlet limit. Such a correction can be order one and make the EW vacuum completely stable rather than metastable. The requisite Higgs–singlet coupling λhs and the singlet self–coupling λs are allowed to be very small, as long as λ2hs/(4λs) is greater than about 0.015 for a TeV–scale singlet. This situation is practically indistinguishable from the SM at low energies unless the Higgs quartic coupling is measured. We also ﬁnd that Higgs inﬂation is possible in our framework since the quartic coupling remains positive at high energies.
Oleg Lebedev, On Stability of the Electroweak Vacuum and the Higgs Portal, 22/05/2012
... the spectral action includes naturally a dilaton ﬁeld which guarantees the scale invariance of the standard model interactions, and provides a mechanism to generate mass hierarchies. This is in addition to the advantages obtained previously in  which are now well known . There it was shown that all the correct features of the standard model are obtained without any ﬁne tuning, such as uniﬁcation with gravity, uniﬁcation of the three gauge coupling constants and relating the Higgs to the gauge couplings. These results should be taken to support the idea that all the geometric information about the physical space is captured by the knowledge of the Dirac operator of an appropriate noncommutative space.
Ali H. Chamseddine, Alain Connes, Scale Invariance in the Spectral Action, 16/03/2005